$X_{\text{sp}}^+(N)$ (awaiting review)

$X_{\text{sp}}^+(N)$ is the modular curve $X_H$ for $H\le \GL_2(\widehat\Z)$ the inverse image of an extended Cartan subgroup $\begin{pmatrix} * & 0 \\ 0 & * \end{pmatrix} \cup \begin{pmatrix} 0 & * \\ * & 0 \end{pmatrix}\subseteq \GL_2(\Z/N\Z)$ that is split at every prime dividing $N$. As a moduli space it parameterizes pairs $(E,\{C,D\})$ where:

• $E$ is an elliptic curve over $k$, and
• $\{C,D\}$ is a $\Gal_k$-stable pair of cyclic subgroups such that $E[N](\overline{k})\simeq C \oplus D$. (Neither $C$ nor $D$ need be $\Gal_k$-stable.)